Page 340 - Introduction to Continuum Mechanics

324 The Elastic Solid

If the functions <PI andy>2 are derived from a potential function/! of the invariants l\ and
/2 such that

then the constitutive equation becomes

and the solid is known as an incompressible hyperelastic isotropic solid.

5.35 Simple Extension of an Incompressible Isotropic Elastic Solid

A rectangular bar is pulled in the x\ direction. At equilibrium, the ratio of the deformed
length to the undeformed length (i.e., the stretch) is AJ in the jq direction and A 2 in the
transverse direction. Thus, the equilibrium configuration is given by

*7
where the condition Aj KI— \ describes the isoehoiic condition (i.e., no change in volume).
The left Cauchy-Green deformation tensor B and its inverse are given by

From the constitutive equation

we have